Introduce Join-Based Algorithms in Batch Operations

[devarajabc]

The current implementation of batch operations converts a sorted array into a perfectly balanced binary tree and colors all nodes black.
However, this approach is limited because:

1. It can only handle node counts of the form $2^n - 1$.
2. It cannot perform batch operations on a non-empty tree. (unjoinable)

To address these limitations, I propose introducing join-based algorithms for batch operations, inspired by Wikipedia’s red–black tree parallel algorithms section.

This involves introducing two new helper operations: rb_split and rb_join.

Ref:

A Verified Cost Analysis of Joinable Red-Black Trees :https://arxiv.org/pdf/2309.11056
Joinable Parallel Balanced Binary Trees: https://dl.acm.org/doi/pdf/10.1145/3512769

[devarajabc]

Proposed Bulk Operations Workflow

Find:

1. Search for the node in the buffer if the buffer size is less than the current tree height.
2. Join the buffer into the tree if the buffer size is larger than the current tree height.
3. Search for the node in the original tree.

Insert:

1. Insert the node into the buffer.
2. Join the buffer into the tree if the buffer is full.

Delete:

1. Search and delete node in the buffer if the buffer size is less than the current tree height.
2. Join the buffer into the tree if the buffer size is larger than the current tree height.
3. Delete the node from the original tree.
4. Join the buffer into the tree.

TODO

Add the time complexity